About vector space and subspaces

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Discussion Overview

The discussion revolves around the concepts of vector spaces and subspaces in linear algebra. Participants explore how to distinguish between these two subjects, focusing on definitions and examples to clarify their understanding.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the distinction between vector spaces and subspaces, noting that subspaces must pass through the origin.
  • Another participant explains that a vector subspace is a vector space in its own right, emphasizing that the term "sub" indicates it is a subset of a larger vector space, both defined over the same field.
  • This participant provides an example of a subspace, stating that the set {t(1,0); t ∈ ℝ} is a subspace of ℝ², while ℝ² is not a subspace of ℝ³ because it is not a subset of ℝ³.
  • A third participant agrees with the previous explanation but finds the example confusing, clarifying that ℝ² and ℝ³ consist of vectors with different dimensions and thus cannot be subsets of one another.
  • This participant introduces the idea that in practice, ℝ² can be viewed as a subset of ℝ³ by appending a zero to the vectors, suggesting that this is a common intuitive understanding.
  • The third participant emphasizes the importance of making explicit the reasoning behind this intuitive view, noting that it contrasts with the earlier example provided.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between ℝ² and ℝ³, with some supporting the idea of treating ℝ² as a subset of ℝ³ under certain conditions, while others maintain that they are fundamentally different due to their dimensionality. The discussion remains unresolved regarding the implications of this intuitive understanding.

Contextual Notes

There is a lack of consensus on the treatment of vector spaces and subspaces, particularly regarding dimensionality and the implications of viewing one as a subset of the other. The discussion also highlights the potential for confusion in examples used to illustrate these concepts.

applechu
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Hi:
I am a newbie to linear algebra; I have a problem about
vector space and subspaces. How to distinguish these two
subject. what I know from books is subspace is going through
zero, but I still can not figure out what is the difference between
vector space and subspaces, thanks.
 
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applechu said:
Hi:
I am a newbie to linear algebra; I have a problem about
vector space and subspaces. How to distinguish these two
subject. what I know from books is subspace is going through
zero, but I still can not figure out what is the difference between
vector space and subspaces, thanks.



Simple: a vector subspace is a vector space in its own, but the particle "sub" indicates that it is a subset of a vector space that

contains it, and they both are vectors spaces wrt the very same operations and, of course, over the same field.

Thus, for example, the set [itex]\,\,\{t(1,0) \,;\,t\in\mathbb{R}\}\,[/itex] is a v. subspace of [itex]\,\mathbb{R}^2\,[/itex], but [itex]\,\mathbb{R}^2\,[/itex] is

not a subspace of [itex]\,\mathbb{R}^3[/itex] is the former as not a subset of the latter.

DonAntonio
 
DonAntonio is correct, but I think his example is confusing.

[itex]\mathbb{R}^2[/itex] is not a subspace of [itex]\mathbb{R}^3[/itex] because the former are vectors with two entries, and the latter are vectors with three entries. They are just different animals altogether, and so cannot be subsets of one or another.

However, in practice we tend to think of [itex]\mathbb{R}^2[/itex] as a subset of [itex]\mathbb{R}^3[/itex] with the tacit understanding that every vector in [itex]\mathbb{R}^2[/itex] gets a zero appended to it.

So [itex][4,2]\in\mathbb{R^2}[/itex] becomes [itex][4,2,0]\in\mathbb{R^3}[/itex]

With this understanding, [itex]\mathbb{R}^2[/itex] is a subspace of [itex]\mathbb{R}^3[/itex]

I add this not to nit pick or be confusing. I add this because this is how most people see [itex]\mathbb{R}^2[/itex] and it's important to make explicit why it is wrong to do so (like in DonAntonio's post) and what we implicitly do in our intuition.
 
Last edited:
Thanks a lot
 

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